SAS Data Analysis Examples: Tobit Models



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SAS Data Analysis Examples
Tobit Analysis

Examples of Tobit Analysis

Example 1. In the 1980s there was a federal law restricting speedometer readings to no more than 85 mph. So if you wanted to try and predict a vehicle's top-speed from a combination of horse-power and engine size, you would get a reading no higher than 85, regardless of how fast the vehicle was really traveling. This is a classic case of right-censoring (censoring from above) of the data. The only thing we are certain of is that those vehicles were traveling at least 85 mph. Tobit models are designed to make improved estimates when there is either left- or right-censoring.

Example 2. A research project is studying the level of lead in home drinking water as a function of the age of a house and family income. The water testing kit cannot detect lead concentrations below 5 parts per billion (ppb). The EPA considers levels above 15 ppb to be dangerous. These data are an example of left-censoring (censoring from below) and can be analyzed using tobit analysis.

Example 3. Consider the situation in which we have a measure of academic aptitude (scaled 200-800) which we want to model using reading and math test scores and whether the student is enrolled in a public or private school. The problem here is that students who answer all questions on the academic aptitude test correctly receive a score of 800, even though it is likely that these students are not "truly" equal in aptitude.

Description of the Data

Let's pursue Example 3 from above.

We have a hypothetical data file, tobitex.sas7bdat , with 200 observations. The academic aptitude variable is apt, the reading and writing test scores are read and write, respectively. The variable public is a zero-one variable with the ones indicating a public school student.

Let's look at the data.

proc means data = tobitex;
run;
	     The MEANS Procedure

Variable      N            Mean         Std Dev         Minimum         Maximum
-------------------------------------------------------------------------------
ID          200     100.5000000      57.8791845       1.0000000     200.0000000
APT         200     651.0600000     101.4403625     420.0000000     800.0000000
READ        200      52.2300000      10.2529368      28.0000000      76.0000000
MATH        200      52.6450000       9.3684478      33.0000000      75.0000000
PUBLIC      200       0.5450000       0.4992205               0       1.0000000
-------------------------------------------------------------------------------
proc univariate data = tobitex plot;
var apt;
histogram / normal;
run;
	     The UNIVARIATE Procedure
Variable:  APT

                            Moments

N                         200    Sum Weights                200
Mean                   651.06    Sum Observations        130212
Std Deviation      101.440362    Variance            10290.1471
Skewness           -0.4439105    Kurtosis            -0.7559199
Uncorrected SS       86823564    Corrected SS        2047739.28
Coeff Variation    15.5808009    Std Error Mean      7.17291682

              Basic Statistical Measures

    Location                    Variability

Mean     651.0600     Std Deviation          101.44036
Median   663.0000     Variance                   10290
Mode     717.0000     Range                  380.00000
                      Interquartile Range    153.00000

	     < some output omitted to save space >

	             Extreme Observations

----Lowest----        ----Highest---

Value      Obs        Value      Obs

  420      153          800       92
  420      133          800       93
  420      126          800      137
  420       16          800      160
  441      128          800      192

   Stem Leaf                          #  Boxplot
     80 000000000000000              15     |
     78 00000000                      8     |
     76                                     |
     74 7777777777777777779999       22     |
     72 66668888                      8  +-----+
     70 7777777777777777777777777    25  |     |
     68 666666666666                 12  |     |
     66 33333333333333333222         20  *-----*
     64 2222222222222221             16  |  +  |
     62 11                            2  |     |
     60 99999999999                  11  |     |
     58 88                            2  |     |
     56 7999999999                   10  +-----+
     54 6888888888888                13     |
     52 555555555577                 12     |
     50 44444666                      8     |
     48 3335                          4     |
     46 2244                          4     |
     44 1111                          4     |
     42 0000                          4     |
        ----+----+----+----+----+
    Multiply Stem.Leaf by 10**+1
	                            Normal Probability Plot
     810+                                        ******** **
        |                                     ***
        |                                    ++
        |                                 *****
        |                               **+
        |                            ****
        |                          ***+
        |                        ***+
        |                      ***+
        |                     **+
        |                    **
        |                  ++*
        |                 +**
        |               +***
        |             ****
        |           ***
        |         +**
        |       +**
        |     ***
     430+** **
         +----+----+----+----+----+----+----+----+----+----+

proc freq data = tobitex;
tables public;
run;
	     The FREQ Procedure

                                   Cumulative    Cumulative
PUBLIC    Frequency     Percent     Frequency      Percent
-----------------------------------------------------------
     0          91       45.50            91        45.50
     1         109       54.50           200       100.00
	proc corr data = tobitex nosimple;
var read math public apt;
run;
	     The CORR Procedure

   4  Variables:    READ     MATH     PUBLIC   APT

          Pearson Correlation Coefficients, N = 200
                  Prob > |r| under H0: Rho=0

                READ          MATH        PUBLIC           APT

READ         1.00000       0.66228      -0.05308       0.59713
                            <.0001        0.4553        <.0001

MATH         0.66228       1.00000      -0.02934       0.61705
              <.0001                      0.6801        <.0001

PUBLIC      -0.05308      -0.02934       1.00000       0.25665
              0.4553        0.6801                      0.0002

APT          0.59713       0.61705       0.25665       1.00000
              <.0001        <.0001        0.0002
	
proc insight data = tobitex;
scatter read math apt * read math apt;
run;

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a tobit analysis, let's consider some other methods that you might use.

OLS Regression - You could analyze these data using OLS regression. OLS regression will treat the 800 as the actual values and not as the lower limit of the top academic aptitude. If we compare the predicted values from an OLS model with a tobit model we would find that the tobit predicted values have a range of 491.33-840.12 while the OLS predicted have a range of 497.76-825.22.
Truncated Regression - Beginners often confuse the concepts of truncation with censoring. With censored data we have all of the observations but we don't know the "true" values of some of them. With truncation some of the observations are not included in the analysis because of the value of the variable. It would be inappropriate to analyze the data in our example using a truncated regression model.

SAS Tobit Analysis

proc qlim data = tobitex ;
model apt = read math public;
endogenous apt ~ censored (ub=800);
output out = temp1 predicted;
run;
	         The QLIM Procedure

                 Summary Statistics of Continuous Responses

                                                                  N Obs N Obs
                      Standard                     Lower    Upper Lower Upper
Variable     Mean        Error       Type          Bound    Bound Bound Bound

  apt      651.06   101.440362     Censored                   800          15

             Model Fit Summary

Number of Endogenous Variables             1
Endogenous Variable                      apt
Number of Observations                   200
Log Likelihood                         -1072
Maximum Absolute Gradient         1.43701E-7
Number of Iterations                      17
AIC                                     2154
Schwarz Criterion                       2171

Algorithm converged.

                      Parameter Estimates

                                 Standard                 Approx
Parameter        Estimate           Error    t Value    Pr > |t|

Intercept      188.394294       32.751473       5.75      <.0001
READ             3.681712        0.687378       5.36      <.0001
MATH             4.557839        0.753892       6.05      <.0001
PUBLIC          62.163301       10.574065       5.88      <.0001
_Sigma          73.632442        3.874453      19.00      <.0001

The ub = option on the endogenous statement indicates the value at which the right-censoring begins. There is also a lb() option to indicate the value of the left-censoring, which was not needed in this example.

At the top, the output provides a summary of the number of left- and right-censored values. In the table entitled Parameter Estimates, we have the tobit regression coefficients, the standard error of the coefficients, the t-Value and the associated p-values. The ancillary statistic _sigma is equivalent to the standard error of estimate in OLS regression. The value of 73.63 can be compared to the standard deviation of academic aptitude, which was 101.44. This shows a substantial reduction. The output also contains an estimate of the standard error of _sigma, as well as the t-Value and p-value. That _sigma is statistically significant means that 73.63 is statistically significantly different from 0. The validity of this test of _sigma is a matter of debate among statisticians, and some programs will produce the estimate and standard error, but not the test of statistical significance.

To get an idea about model fit, you can use the squared multiple correlation between the outcome variable (apt) and the predicted value. The predicted values are obtained by creating an output data set (which we called temp1) on the output statement with the predict option. The following proc corr and data step are then used to get the desired value.

proc qlim data=tobitex ;
model apt = read math public;
endogenous apt ~ censored (ub=800);
output out = temp1 predicted;
run;

ods output PearsonCorr=tobit_corr;
proc corr data = temp1 nosimple;
var apt p_apt;
run;

data _null_;
set tobit_corr;
if variable = "APT";
file print;
a = round((p_apt)**2, .0001);
put "The squared multiple correlation between apt and the predicted value is " a;
run;

The squared multiple correlation between apt and the predicted value is 0.527

Sample Write-Up of the Analysis

In the tobit regression model predicting academic aptitude from reading, math and public school, each of the predictor variables in the model was also statically significant at the .001 level. The squared correlation between the observed and predicted academic aptitude values was 0.53, indicating that these three predictors accounted for over 50% of the variability in the outcome variable. A unit change in read and math lead to a 3.68 and 4.56 increase in the predicted aptitude, respectively. Attending a public school increased the predicted aptitude by 62.16 points as compared with private school attendance.